Publ. RIMS, Kyoto Univ.
42 (2006), 787–802
On the Occupation Time on the Half Line
of Pinned Diffusion Processes
By
Yuko Yano∗
Abstract
The aim of the present paper is to generalize Lévy’s result of the occupation
time on the half line of pinned Brownian motion for pinned diffusion processes. An
asymptotic behavior of the distribution function at the origin of the occupation time
Γ+ (t) and limit theorem for the law of the fraction Γ+ (t)/t when t → ∞ are studied.
An expression of the distribution function by the Riemann–Liouville fractional integral
for pinned skew Bessel diffusion processes is also obtained. Krein’s spectral theory
and Tauberian theorem play important roles in the proofs.
§0.
Introduction
P. Lévy’s arc-sine law is a well-known result for a Brownian motion: Let
t
B = {Bt , Px } be a standard Brownian motion on R1 and Γ+ (t) = 0 1[0,∞)
(Bs )ds, i.e., the occupation time on [0, ∞). Then, for each t > 0, we have
1
√
2
(0.1)
P0 Γ+ (t) ≤ x = P0 (Γ+ (1) ≤ x) = arcsin x, 0 ≤ x ≤ 1
t
π
and
(0.2)
P0
1
t
Γ+ (t) ≤ x Bt = 0 = P0 (Γ+ (1) ≤ x|B1 = 0) = x,
0 ≤ x ≤ 1.
Many authors have been interested in these results and tried to extend them
for more general stochastic processes. In the present paper we shall confine
Communicated by Y. Takahashi. Received December 17, 2004. Revised April 14, 2005.
2000 Mathematics Subject Classification(s): Primary 60J25; Secondary 60J60, 60F99.
Key words: Brownian motion, arc-sine law, speed measure, Krein’s theory, Tauberian
theorem
∗ Department of Information Sciences, Ochanomizu University, 2-1-1 Otsuka Bunkyo-ku
Tokyo 112-8610, Japan.
e-mail: yano@is.ocha.ac.jp
c 2006 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
788
Yuko Yano
ourselves to the case of one-dimensional diffusion processes and we are interested in the following two results: The first one is due to S. Watanabe [10].
He proved that all possible limiting laws of the fraction Γ+ (t)/t as t → ∞ are
Lamperti’s laws (3.1), which can be identified with the laws of the the time
spent on the positive side by skew Bessel diffusion processes. The second one is
due to Y. Kasahara and the author [5], which studies the relationship between
the asymptotic behavior of the distribution function of Γ+ (t) and that of the
speed measure at x = 0.
However, we notice that these results are only concerned with (0.1) and
it might be natural to study similar problems for (0.2). Thus the aim of this
article is to study the same problems as above for pinned diffusion processes.
First, we shall determine the asymptotic behavior of the distribution function
at x = 0, which explains how it depends on the positive and negative sides
of the speed measure (Theorem 2.1). Secondly, we shall obtain an expression
of the distribution function by the Riemann–Liouville fractional integral for
pinned skew Bessel diffusion processes (Theorem 3.1). Finally, we shall study
a limit theorem for pinned diffusion processes (Theorem 3.2).
The idea of the proofs for the pinned diffusion processes are essentially the
same as for non-pinned cases. The key to our proofs is the double Laplace transform formula (2.4) and Tauberian theorems for Laplace transform. However, it
should be emphasized here that the proofs cannot be carried out completely in
parallel. The difficulty is that we cannot apply the continuity theorem directly,
because, unlike the non-pinned case, the index of the regular variation of the
functions appearing in the double Laplace transform (2.4) is out of the range.
In Section 1, we shall introduce some notations and a brief review of Krein’s
correspondence and the generalized diffusion processes. Although they might be
cumbersome to readers who are familiar to these materials, they are necessary
to state and to prove our results. We shall state the asymptotic result with
the proof in Section 3, where the double Laplace transform formula is also
proved. Section 4 is devoted to our limit theorem. In Appendix, we shall give a
simple version of the inversion formula for the (generalized) Stieltjes transform
of arbitrary order.
§1.
Preliminaries
We adopt the same notation as in [5] but we state it here for the completeness of the paper. See [6] and [4] for details.
Let m : [0, l) → [0, ∞) be a right-continuous, nondecreasing function where
0 ≤ l ≤ ∞. We put m(0−) = 0 and m(x) = ∞ for x ≥ l when l < ∞ so that the
The Occupation Time of Pinned Diffusions
789
Borel measure dm is defined on [0, l). Such dm is referred to as an inextensible
measure. Let M be the class of all such functions m. m ∈ M is sometimes
called a string. To a given m ∈ M, we assign a function h(λ) defined on (0, ∞)
in the following way: If m(x) ≡ ∞, then h(λ) ≡ 0. Otherwise, for λ > 0, we
consider the following integral equations:
x ξ+
φ(x, λ) = 1 + λ
dξ
φ(u, λ)dm(u),
0
ψ(x, λ) = x + λ
0−
ξ+
x
dξ
0
ψ(u, λ)dm(u)
0−
on the interval [0, l). The equations have unique continuous solutions φ(·, λ)
and ψ(·, λ) on [0, l) for each λ > 0. We then define
h(λ) = lim
x↑l
ψ(x, λ)
=
φ(x, λ)
l
0
dx
.
φ(x, λ)2
We note that, if m(x) = 0 for 0 ≤ x < l, then h(λ) ≡ l. The correspondence
m ∈ M → h(λ) is called Krein’s correspondence. h(λ) is called the spectral
characteristic function of the string m.
The spectral characteristic function h(λ), h ≡ ∞, has a unique representation
dσ(ξ)
h(λ) = c +
(1.1)
[0,∞) λ + ξ
for some 0 ≤ c < ∞ and nonnegative Radon measure dσ on [0, ∞) such that
dσ(ξ)
< ∞.
[0,∞) 1 + ξ
The measure dσ is called the spectral measure and the function σ(t) = [0,t] dσ(ξ)
is called the spectral function associated with the string m. In fact, it holds that
(1.2)
lim h(λ) = c = inf{x ≥ 0; m(x) > 0} = inf Supp(dm).
λ→∞
Let H be the class of all functions h of the form (1.1) and h ≡ ∞. An important
result is that Krein’s correspondence
m ∈ M → h ∈ H
is one-to-one and onto and defines a homeomorphism if suitable topologies are
introduced on M and H.
790
Yuko Yano
Let m∗ := m−1 ∈ M be the right-continuous inverse of m ∈ M. If
m ←→ h is Krein’s correspondence, then
dσ ∗ (ξ) 1
m∗ (x) ←→
(1.3)
=: h∗ (λ) = c∗ +
λh(λ)
[0,∞) λ + ξ
is also Krein’s correspondence. The function m∗ is called the dual string of m.
Put ψ = 1/h. The function ψ is called the spectral characteristic exponent
of the string m. We remark that ψ(λ), λ > 0, is known to have the form
∞
ψ(λ) = c0 + c1 λ +
(1.4)
(1 − e−λu )n(u)du
0
∗
where c0 = 1/l, c1 = c and
e−ξu ξdσ ∗ (ξ).
n(u) =
(0,∞)
Now let m+ , m− ∈ M such that
m± : [0, l± ) → [0, ∞)
and m− (0) = 0. We define a Radon measure dm(x) on (−l− , l+ ) by
dm+ (x) on [0, l+ ),
dm(x) =
dm̌− (x) on (−l− , 0)
where dm̌− (x) is the image measure of dm− under the mapping x → −x.
Then a strong Markov process X = {Xt , Px } on Em = (Supp(dm) ∪ {−l− } ∪
d
d
{l+ }) ∩ (−∞, ∞) associated with the Feller generator A = dm(x)
dx can be
1
constructed from the Brownian motion B on R by the time change. Here
the boundaries l+ and −l− are traps for the Markov process X. This process
is called the generalized diffusion process corresponding to the pair of strings
{m+ , m− }. We denote the transition probability density of X with respect to
dm as p(t, x, y), i.e.,
Px (Xt ∈ dy) = p(t, x, y)dm(y).
We remark that
(1.5)
0
∞
e−λt p(t, 0, 0)dt =
1
,
ψ+ (λ) + ψ− (λ)
λ > 0.
Let X = {Xt , Px } be a generalized diffusion process on (−l− , l+ ) corresponding to the pair {m+ , m− } so that m± ∈ M with m− (0) = 0 and let h±
The Occupation Time of Pinned Diffusions
791
be the spectral characteristic functions and ψ± be the spectral characteristic
exponents associated with strings m± , respectively. We call a process X the
skew Bessel diffusion process of dimension 2 − 2α, 0 < α < 1, with the skew parameter p, 0 ≤ p ≤ 1, and denote it as SKEW BES(2 − 2α, p) if it corresponds
to the pair {m+ , m− } given by
m+ (x) = p1/α x1/α−1 ,
m− (x) = (1 − p)
l+ = ∞,
1/α 1/α−1
x
,
l− = ∞.
We remark that, if p = 0 or p = 1, the process X is the Bessel process in its
canonical scale which is reflecting at 0. The conditions for m± is equivalent to
the following:
h+ (λ) = Dα p−1 λ−α ,
h− (λ) = Dα (1 − p)−1 λ−α
and
ψ+ (λ) = pDα−1 λα ,
ψ− (λ) = (1 − p)Dα−1 λα
where Dα = {α(1 − α)}−α Γ(1 + α)/Γ(1 − α). When p = α = 1/2, it is a
Brownian motion with a multiplicative constant.
§2.
Asymptotic Behavior of the Occupation Time Distribution
Function
Our main result of this section is the following:
Theorem 2.1.
Let X = {Xt , Px } be a generalized diffusion process on
t
(−∞, ∞) corresponding to {m+ , m− } and let Γ+ (t) = 0 1[0,∞) (Xs )ds. Let
ϕ(x) vary regularly at 0 with exponent 1/α, 0 < α < 1. If
(2.1)
ϕ(x)
,
x
m+ (x) ∼
x → 0+,
then
(2.2) P0 (Γ+ (t) ≤ x|Xt = 0) ∼
− d g ∗ (t)
Dα2
· dt − {ϕ−1 (x)}2 ,
Γ(1 + 2α) p(t, 0, 0)
where
(2.3)
∗
g−
(t) =
[0,∞)
x→0+
∗
e−tξ dσ−
(ξ).
792
Yuko Yano
Remark. If ϕ(x) is a regularly varying function at 0 with exponent 1/α,
then the asymptotic inverse ϕ−1 (y) is defined and varies regularly with exponent α.
∗
Remark. g−
(t) is equal to the transition density p∗− (t, 0, 0) of a diffusion
process on (−∞, 0] corresponding to {0, m∗− } (0 is a reflecting barrier).
To prove Theorem 2.1, we prepare the following formula:
Theorem 2.2.
Let X = {Xt , Px } be a generalized diffusion process cort
responding to {m+ , m− } on (−l− , l+ ) and Γ+ (t) = 0 1[0,∞) (Xs )ds. Let ψ± be
the characteristic exponents of m± , respectively. Then, for λ > 0, µ > 0,
∞
1
(2.4)
.
e−µt E0 [e−λΓ+ (t) |Xt = 0]p(t, 0, 0)dt =
ψ
(λ
+
µ)
+ ψ− (µ)
+
0
Proof. Put
zε (x) = Ex
0
∞
1[0,ε) (Xt ) −µt−λΓ+ (t) e
dt
m[0, ε)
for λ > 0, µ > 0. Then, according to Feynman–Kac formula, the function zε (x)
satisfies the equation
1[0,ε) (x)
d d
µ + λ · 1[0,∞) −
zε (x) =
.
dm(x) dx
m[0, ε)
The value at x = 0 of the unique bounded solution of the preceding equation is
1
1
u2 (y, λ + µ)dm(y)
zε (0) =
m[0, ε) ψ+ (λ + µ) + ψ− (µ) [0,ε)
where u2 is the positive non-increasing solution of (µ + λ −
with u(0) = 1. Then, letting ε → 0, we have
zε (0) →
d
d
dm(x) dx )u(x)
=0
1
.
ψ+ (λ + µ) + ψ− (µ)
On the other hand,
(2.5)
∞ 1
[0,ε) (Xt ) −µt−λΓ+ (t)
zε (0) = E0
e
dt
m[0, ε)
0
∞
1
[0,ε) (Xt ) −λΓ+ (t)
dt
e
e−µt E0
=
m[0, ε)
0
∞
1
=
e−µt
E0 e−λΓ+ (t) |Xt = y p(t, 0, y)dm(y).
m[0, ε) [0,ε)
0
793
The Occupation Time of Pinned Diffusions
Now let
p̃(t, x, y) := Ex e−λΓ+ (t) |Xt = y p(t, x, y).
Then p̃(t, x, y) is the transition density of the diffusion process with generator
d
d
à = dm(x)
dx −λ·1[0,∞) . It is shown by McKean [8] that p̃(t, x, y), as a function
of y, belongs to the domain of Ã, in particular, it is continuous in y. Hence
∞
the RHS of (2.5) →
e−µt E0 [e−λΓ+ (t) |Xt = 0]p(t, 0, 0)dt, ε → 0+.
0
Therefore we obtain
∞
e−µt E0 [e−λΓ+ (t) |Xt = 0]p(t, 0, 0)dt =
0
1
.
ψ+ (λ + µ) + ψ− (µ)
Now we give the proof to our theorem.
Proof. By Theorem 2.2, we see
∞
e−µt E0 [e−λΓ+ (t) |Xt = 0]p(t, 0, 0)dt =
0
1
ψ+ (λ + µ) + ψ− (µ)
for λ > 0, µ > 0. Differentiating the both sides with respect to µ, we have
∞
∂
∂µ ψ+ (λ + µ) + ψ− (µ)
−µt
−λΓ+ (t)
(2.6)
e E0 [e
|Xt = 0]p(t, 0, 0)tdt = 2 .
0
ψ+ (λ + µ) + ψ− (µ)
We note by [4] that (2.1) is equivalent to
h+ (λ) =
(2.7)
1
1
∼ Dα ϕ−1
,
ψ+ (λ)
λ
λ → ∞.
Thus, ψ+ (λ) varies regularly at ∞ with exponent 0 < α < 1 and hence
∂
∂µ ψ+ (λ + µ) → 0 as λ → ∞. Therefore, as λ → ∞,
the RHS of (2.6) ∼
and we obtain
2
(2.8) ψ+ (λ)
0
∞
d
2
(λ)
ψ− (µ) ψ+
dµ
e−µt E0 [e−λΓ+ (t) |Xt = 0]p(t, 0, 0)tdt →
d
ψ− (µ),
dµ
λ → ∞.
794
Yuko Yano
On the other hand, by (1.3), 1/(µh− (µ)) = ψ− (µ)/µ is the characteristic
function of the dual string m∗− and hence there exists a nonnegative Radon
∞ ∗
∗
measure dσ−
on [0, ∞) such that 0 dσ−
(ξ)/(1 + ξ) < ∞ and
∞ ∗
∞
∞
dσ− (ξ)
1
ψ− (µ)
−µt
∗
=
=
=
e dt
e−tξ dσ−
(ξ)
µ
µh− (µ)
µ+ξ
0
0
0
∞
∗
e−µt g−
(t)dt.
=
0
c∗−
= 0 by m− (0) = 0 and (1.2). Therefore, we have
∞
d
d
∗
e−µt − g−
(t) tdt.
ψ− (µ) =
dµ
dt
0
We note that the constant
Since p̃(t, 0, 0) = E0 [e−λΓ+ (t) |Xt = 0]p(t, 0, 0) is monotone in t, (2.8) implies
2
ψ+
(λ)E0 [e−λΓ+ (t) |Xt = 0]p(t, 0, 0) → −
d ∗
g (t),
dt −
λ→∞
by the continuity lemma (see Lemma 2 of [5]). Therefore
E0 [e−λΓ+ (t) |Xt = 0] ∼
d ∗
g− (t)
− dt
1
·
,
2
ψ+ (λ) p(t, 0, 0)
λ → ∞.
By (2.7) and Karamata’s Tauberian theorem, we have
P0 (Γ+ (t) ≤ x|Xt = 0) ∼
− d g ∗ (t)
Dα2
· dt − {ϕ−1 (x)}2 ,
Γ(1 + 2α) p(t, 0, 0)
x → 0+
which completes the proof of the theorem.
Example 1.
0 < p < 1, we put
(2.9)
When X = {Xt , Px } is SKEW BES(2−2α, p), 0 < α < 1,
Gα,p (x) := P0 (Γ+ (1) ≤ x|X1 = 0) = P0
1
t
Γ+ (t) ≤ x Xt = 0 .
As is shown in the proof of Theorem 3.1 below, Gα,p (x) is characterized by
1
dGp,α (x)
1
(2.10)
=
.
α
α
p(1 + ξ) + (1 − p)ξ α
0 (ξ + x)
By inverting this, we obtain an expression (3.2) by the Riemann–Liouville fractional integral as given below. When α = 1/2, that is, the case of skew Brownian motions, we have a more explicit formula: G 12 ,p (x) has the density g 12 ,p (x)
given by
(2.11)
g 12 ,p (x) =
3
p(1 − p)
{(1 − 2p)x + p2 }− 2 ,
2
0 < x < 1.
The Occupation Time of Pinned Diffusions
795
It is readily verified by an elementary calculus that this satisfies (2.10). In
particular, G 12 , 12 (x) = x, which is just Lévy’s result (0.2).
Theorem 2.1 implies that the asymptotic behavior of Gα,p (x) is as follows:
Gα,p (x) ∼
1−p
Γ(1 + α)
· 2 x2α ,
Γ(1 + 2α)Γ(1 − α)
p
x → 0+.
This can be obtained from the expression (3.2) given as below.
§3.
Limit Theorem
In the case of non-pinned SKEW BES(2 − 2α, p), 0 < α < 1, 0 < p < 1,
the density formula of the fraction of the occupation time is given by the form
(3.1)
fα,p (x) =
p(1 − p)xα−1 (1 − x)α−1
sin απ
π p2 (1 − x)2α + (1 − p)2 x2α + 2p(1 − p)xα (1 − x)α cos απ
for 0 < x < 1 (see [7] and [10]). This is Lamperti’s density formula. In the
pinned SKEW BES(2−2α, p) case, we can determine the distribution function
of the fraction Γ+ (t)/t, i.e., Gα,p (x), 0 < α < 1, 0 < p < 1, by inverting double
Laplace transform formula (2.10).
For 0 ≤ x ≤ 1,
Theorem 3.1.
(3.2)
Gα,p (x) =
sin απ
π
0
x
(1 − p)(x − s)α−1 sα ds
.
p2 (1 − s)2α + (1 − p)2 s2α + 2p(1 − p)sα (1 − s)α cos απ
Proof. Applying Theorem 2.2, we have
∞
Dα α−1
Dα
t
(3.3)
e−µt E0 [e−λΓ+ (t) |Xt = 0]
dt =
α
Γ(α)
p(λ + µ) + (1 − p)µα
0
for λ > 0, µ > 0 in the SKEW BES(2 − 2α, p) case. By the self-similarity of
d
the Bessel process, i.e., (Γ+ (t), Xt ) = (tΓ+ (1), tα X1 ), we have
∞
Dα α−1
t
the LHS of (3.3) =
e−µt E0 [e−λtΓ+ (1) |X1 = 0]
dt
Γ(α)
0
∞
Dα
E0
e−{µ+λΓ+ (1)}t tα−1 dt X1 = 0
=
Γ(α)
0
1
= Dα E0
X1 = 0
α
{µ + λΓ+ (1)}
1
dGp,α (x)
.
= Dα
(µ
+ λx)α
0
796
Yuko Yano
Setting ξ = µ/λ(> 0), we obtain
(2.10)
0
1
dGp,α (x)
1
=
.
α
α
(ξ + x)
p(1 + ξ) + (1 − p)ξ α
By the inversion formula for the Stieltjes transform (see Appendix or [9]), we
obtain the equality (3.2) and complete the proof.
Now we introduce a class of random variables {ξα,p }, 0 < α ≤ 1, 0 ≤ p ≤
1, as follows: Each ξα,p is a [0, 1]-valued random variable with the Stieltjes
transform of order α given by
(3.4)
E
1
1
,
=
α
α
(λ + ξα,p )
p(1 + λ) + (1 − p)λα
λ > 0.
When 0 < α < 1, the distribution function of ξα,p is given by Gα,p (x). If α = 1,
ξ1,p is the constant random variable such that P [ξ1,p = p] = 1. If p = 0 or
p = 1, ξα,0 = 0 a.s. and ξα,1 = 1 a.s.
We now state the limit theorem for the law of Γ+ (t)/t as t → ∞ for the
pinned diffusion processes. Let X = {Xt } be a generalized diffusion process
corresponding to {m+ , m− }. We always assume that X0 = 0, that is, we
consider the probability law P0 unless otherwise stated.
Theorem 3.2.
Let Γ+ (t) =
t
0
1[0,∞) (Xs )ds. If
m± (x) ∼ x α −1 K± (x),
1
(3.5)
x→∞
for 0 < α ≤ 1 where K± (x) are slowly varying functions at ∞ with
1
pα
K+ (x)
=
1 ,
x→∞ K− (x)
(1 − p) α
lim
(3.6)
then the distribution of Γ+ (t)/t conditional on Xt = 0 converges in law to that
of ξα,p as t → ∞ i.e.,
P
Remark.
(see [4]):
(3.7)
1
t
Γ+ (t) ≤ x Xt = 0 → P (ξα,p ≤ x),
t → ∞.
The condition (3.5) with (3.6) is equivalent to the following
h± (λ) =
1
1
,
∼ Dα λ−α L±
ψ±
λ
λ→0+
The Occupation Time of Pinned Diffusions
797
where L± (λ) are slowly varying functions at ∞ with
(3.8)
lim
λ→∞
and
p
L+ (λ)
=
L− (λ)
1−p
Dα =
1,
{α(1 − α)}−α Γ(1 + α)/Γ(1 − α),
α = 1,
0 < α < 1.
We prove Theorem 3.2. By (1.5), for µ > 0 and β > 0, we have
∞
∞
−µ
t
β
e
p(t, 0, 0)dt = β
e−µt p(tβ, 0, 0)dt
0
0
=
ψ+ ( µβ )
1
+ ψ− ( βµ )
=
1
1
·
ψ− ( µβ ) ψ+ ( βµ )/ψ− ( µβ ) + 1
∼
1
· (1 − p),
ψ− ( µβ )
∼
1−p
,
ψ− ( β1 )µα
β→∞
β→∞
by the assumption that ψ± vary regularly with α, 0 < α ≤ 1. Hence we have
∞
1 ∞
tα−1
1
1
−µt
· βψ−
dt, β → ∞.
e p(tβ, 0, 0)dt → α =
e−µt
1−p
β 0
µ
Γ(α)
0
Then we obtain
1
1
tα−1
· βψ−
,
· p(tβ, 0, 0) →
1−p
β
Γ(α)
(3.9)
β → ∞.
By Theorem 2.2 and (3.7), we have
1 ∞
λ
1
e−µt E e− β Γ+ (tβ) Xtβ = 0 p(tβ, 0, 0)dt
· βψ−
1−p
β 0
1 ∞ µ λ
1
· ψ−
e− β t E e− β Γ+ (t) Xt = 0 p(t, 0, 0)dt
=
1−p
β 0
1
1
1
=
· ψ−
·
λ+µ
1−p
β
ψ+ ( β ) + ψ− ( βµ )
=
ψ− ( β1 )
1
1
·
µ ·
λ+µ
1 − p ψ− ( β ) ψ+ ( β )/ψ− ( βµ ) + 1
→
1
1
1
1
·
=
·
1 − p µα p(λ + µ)α /(1 − p)µα + 1
p(λ + µ)α + (1 − p)µα
798
Yuko Yano
as β → ∞. By (3.3),
1
=
p(λ + µ)α + (1 − p)µα
∞
e−µt E[e−λtξα,p ]
0
Combining these with (3.9), we obtain that
λ
E e− β Γ+ (tβ) Xtβ = 0 → E[e−λtξα,p ],
and hence
P
1
β
tα−1
dt.
Γ(α)
β→∞
Γ+ (tβ) ≤ x Xtβ = 0 → P (tξα,p ≤ x),
β→∞
for every t > 0. This clearly implies that
1
P Γ+ (t) ≤ x Xt = 0 → P (ξα,p ≤ x) = Gα,p (x),
t
t → ∞,
which completes the proof.
§4.
Appendix: An Inversion Formula for the Stieltjes Transform
First we recall the Riemann–Liouville fractional integral of order ρ > 0:
x−
1
ρ
I [f ](x) :=
(x − ξ)ρ−1 f (ξ)dξ
Γ(ρ) 0+
1
=
1(0,x) (ξ)(x − ξ)ρ−1 f (ξ)dξ.
Γ(ρ)
Obviously, when ρ ≥ 1, the integral I ρ [f ](x) is well-defined and is a continuous
function on [0, ∞).
Lemma 4.1.
Let 0 < ρ < 1 and let f (x) be a Borel measurable function on [0, ∞) which is bounded on each compact subset of [0, ∞). Then the
Riemann–Liouville fractional integral of order ρ of the function f is well-defined
and is a continuous function on (0, ∞).
Proof. It is obvious noting the following: Let ε > 0 such that ε < ρ.
Then there exists a constant C such that
|xρ−1 − y ρ−1 | ≤ C|x − y|ε {xρ−ε−1 + y ρ−ε−1 }
for any x, y > 0.
It is easy to check the following:
The Occupation Time of Pinned Diffusions
799
Lemma 4.2.
Let ρ1 , ρ2 > 0 and let f (x) be a Borel measurable function
on [0, ∞) which is bounded on each compact subset of [0, ∞). Then it holds that
I ρ1 ◦ I ρ2 [f ](x) = I ρ1 +ρ2 [f ](x)
for any x ∈ (0, ∞).
Lemma 4.3.
Let ρ > 1 and let f (x) be a Borel measurable function on
[0, ∞) which is bounded on each compact subset of [0, ∞). Then I ρ [f ](x) has
a continuous derivative given by
d ρ
I [f ](x) = I ρ−1 f (x)
dx
for any x ∈ (0, ∞).
Let F (x) be a non-negative increasing function on [0, ∞). For 0 < ρ < 1
the Stieltjes transform of the Stieltjes measure dF of order ρ is defined by
dF (ξ)
.
Sρ [dF ](x) =
(x
+ ξ)ρ
[0,∞)
If the Riemann–Stieltjes integral of the right hand side converges at a point
x ∈ (0, ∞), then it converges uniformly on any compact subset of C \ (−∞, 0]
and hence the function Sρ [dF ] is a holomorphic function on C \ (−∞, 0].
Theorem 4.1.
Suppose that the Stieltjes transform Sρ [dF ] converge at
a point x ∈ (0, ∞). Then the limit
x−
1
dσ{Sρ [dF ](−σ − iη) − Sρ [dF ](−σ + iη)}
Φ(x) := lim
η→0+ 2πi 0+
exists for any x ∈ (0, ∞) and the equality
Φ(x) =
1 1−ρ
I
[F − F (0)](x)
Γ(ρ)
holds for any x ∈ (0, ∞). That is, the function x → Φ(x) is a continuous
function in x such that
x−
{F (ξ) − F (0)}dξ = Γ(ρ)I ρ [Φ](x)
0+
for any x ∈ (0, ∞). Moreover, suppose the limit
φ(σ) := lim
η→0+
1
{Sρ [dF ](−σ − iη) − Sρ [dF ](−σ + iη)}
2πi
800
Yuko Yano
exist for any σ ∈ (0, ∞) and there exist a function φ̃ such that
|Sρ [dF ](−σ − iη) − Sρ [dF ](−σ + iη)| ≤ φ̃(σ)
and
x−
φ̃(σ)dσ < ∞
0+
for any x ∈ (0, ∞). Then it follows that F (x) is continuous and
F (x) − F (0) = Γ(ρ)I ρ [φ](x)
for any x ∈ (0, ∞).
Proof. Let x > 0 be fixed.
x−
1
dσ{Sρ [dF ](−σ − iη) − Sρ [dF ](−σ + iη)}
2πi 0+
x−
1
=
dσ
{(ξ − σ − iη)−ρ − (ξ − σ + iη)−ρ }dF (ξ)
2πi 0+
[0,∞)
x−
1
=
dσ
{(ξ − σ − iη)−ρ − (ξ − σ + iη)−ρ }dF (ξ)
2πi 0+
[0,x)
x−
1
+
dσ
{(ξ − σ − iη)−ρ − (ξ − σ + iη)−ρ }dF (ξ)
2πi 0+
[x,2x)
x−
1
+
dσ
{(ξ − σ − iη)−ρ − (ξ − σ + iη)−ρ }dF (ξ)
2πi 0+
[2x,∞)
=: S1 + S2 + S3 .
Then we show that
lim S1 =
η→0+
1 1−ρ
I
[F ](x)
Γ(ρ)
and
lim S2 = lim S3 = 0.
η→0+
η→0+
First, we obtain that
x−
1
dF (ξ)
{(ξ − σ − iη)−ρ − (ξ − σ + iη)−ρ }dσ
S1 =
2πi [0,x)
0+
x−
1
=
{F (ξ) − F (0)}{(ξ − x − iη)−ρ − (ξ − x + iη)−ρ }dξ
2πi 0+
sin ρπ x−
{F (ξ) − F (0)}(x − ξ)−ρ dξ
→
π
0+
1 1−ρ
=
I
[F ](x).
Γ(ρ)
The Occupation Time of Pinned Diffusions
801
The convergence is justified by
x−
(4.1)
(x − ξ)−ρ dξ < ∞.
0+
It easily follows that S2 converges to zero from (4.1) and
lim {(x − σ − iη)−ρ − (x − σ + iη)−ρ } = 0
η→0+
if x − σ > 0.
Finally, note that
η
x−
1
S3 =
dσ
dF (ξ)
iρ(ξ − σ − iζ)−ρ−1 dζ.
2πi 0+
[2x,∞)
−η
Thus,
x−
η
|S3 | ≤
dF (ξ)
2ηρ(ξ − σ)−ρ−1 dσ
π [2x,∞)
0+
η
=
{(ξ − x)−ρ − ξ −ρ }dF (ξ)
πρ [2x,∞)
→0
as η → 0+.
In the case of F (x) = Gα,p (x), 0 < α < 1, 0 < p < 1, we have
Sα [dF ](x) =
p(1 +
x)α
1
.
+ (1 − p)xα
Then we can compute φ(σ) as
φ(σ) =
sin απ
(1 − p)σ α
π p2 (1 − σ)2α + (1 − p)2 σ 2α + 2p(1 − p)σ α (1 − σ)α cos απ
and hence obtain (3.2).
Acknowledgments
The author is thankful to Professor Yuji Kasahara for several suggestions
and encouragements and to Professor Yoichiro Takahashi and Professor Shinzo
Watanabe for fruitful discussions. This work was completed while the author
was in the Research Institute for Mathematical Sciences, Kyoto University. She
thanks the Institute for the hospitality.
802
Yuko Yano
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